Weakness of the Topology of a JB*-Algebra

1992 ◽  
Vol 35 (4) ◽  
pp. 449-454 ◽  
Author(s):  
Ali Bensebah
Keyword(s):  
Complete Algebra ◽  
Usual Norm

AbstractThe main purpose of this paper is to prove, that the topology of any (non-complete) algebra norm on a JB* -algebra is stronger than the topology of the usual norm. The proof of this theorem consists of an adaptation of the recent Rodriguez proof [8] that every homomorphism from a complex normed (associative) Q-algebra onto a B*-algebra is continuous.

Automatic continuity with application to C*-algebras

1990 ◽  
Vol 107 (2) ◽  
pp. 345-347 ◽  
Author(s):  
Angel Rodriguez Palacios
Keyword(s):  
Banach Algebras ◽  
Direct Consequence ◽  
Automatic Continuity ◽  
Associative Algebras ◽  
C Algebras ◽  
Complete Algebra ◽  
Usual Norm

The fact proved by Cleveland [4], that the topology of any (non-complete) algebra norm on a C*-algebra is stronger than the topology of the usual norm, is reencountered as a direct consequence of a theorem, which we prove in this note, stating that homomorphisms from certain non-complete normed (associative) algebras onto some semisimple Banach algebras are automatically continuous.

scholarly journals CARLOS POTENCIANO REYES, MD (1940 - 2020) Little-Known but Significant Pioneer

2020 ◽  
Vol 35 (1) ◽  
pp. 83
Author(s):  
Generoso Abes
Keyword(s):  
Operating Room ◽  
Temporal Bone ◽  
First Year ◽  
Sensitivity Index ◽  
Conference Room ◽  
Usual Norm ◽  
Postgraduate Studies ◽  
Training Institutions ◽  
Set Up ◽  
Audiologic Evaluation

Consultants and more senior co-resident physicians at the Philippine General Hospital (PGH) would call him “Caloy.” Hardly would I hear anybody (including our ENT department secretary) address him as Dr. Reyes. This was not because he was not a respected faculty member. Rather, he was everybody’s friend and he probably preferred to be addressed by his nickname. Dr. Carlos P. Reyes was a tall, friendly guy, easily recognizable while walking through the short PGH corridor stretching from the old ENT Ward (Ward 3) to the old ENT operating room (OR) called Floor 15, later designated as the PGH Nursing Office. He would almost always be holding either an expensive photography camera, electronic gadget, ENT OR instrument, or car magazines – suggesting his varied interests aside from having good knowledge of Otolaryngology, particularly Otology. He would usually stop and chat with an acquaintance about his new medical or non-medical interests. I first met Dr. Caloy when I was the first year resident assigned to the Otology section. He would call me “Ging” while presenting the ear patients at the outpatient department (OPD) Ear Clinic, only to learn later that he would address all unfamiliar persons by that name. He was kind, helpful and very understanding. Equipped with ample information in Otology he gathered from postgraduate studies abroad, he would selflessly share these with the residents in order to sharpen our diagnostic acumen. He would instruct us to rely on concise yet complete clinical examination, involving audiologic evaluation tools and meager radiologic information in considering differential diagnoses. He was quite willing to assist us in our learning processes, particularly on how to distinguish middle ear from inner ear disorders, and cochlear versus retrocochlear diseases. Since we did not have any audiologist at that time, he admonished us to carry out the needed audiometric evaluations on our ear patients ourselves in order to learn both the techniques of the procedures and their limitations. Hence, after the OPD clinic we would not only perform routine pure tone and speech audiometric tests but also special examinations like the Bekesy test, short increment sensitivity index (SISI) test, alternate binaural loudness balance (ABLB) test and the test for tone decay. We would then discuss the test results during our next ear clinic and we would listen and be amazed at how Dr. Caloy would integrate the information and arrive at the complex diagnosis. Dr. Caloy was our mentor at the time when refinements in tympanoplasty and mastoidectomy aroused the excitement and imagination of budding otologists worldwide. Whereas canal down mastoidectomy was the usual norm to safely remove common mastoid pathology like cholesteatoma, Dr. Caloy introduced the concept of intact canal wall mastoidectomy that avoids or mitigates recurrent postoperative cleaning of the mastoid bone. The period was also the dawn of neuro-otology when Dr. William House popularized the transmastoid approach for acoustic neuroma and the endolymphatic mastoid shunt as treatment for Meniere’s disease. In order to teach us the anatomical and surgical principles of performing these procedures, Dr. Caloy set up the first temporal bone dissection laboratory in the country at the mezzanine above the ENT conference room. He would offer the course to all ENT residents-in-training and consultants nationwide. He practically revolutionized the method of otologic surgery by requiring ENT surgeons to practice doing ear surgery in the temporal bone dissection lab prior to performing ear surgeries in the operating room. In addition, he advocated the use of the operating microscope and dental drills in place of the old bone gouges, chisels and bone ronguers. His ideas were later adopted by other ENT training institutions as we see today. The requirement that every ENT resident must undergo temporal bone dissection in the course of his training obviously stemmed from the efforts of Dr. Caloy. Many senior ENT consultants who are still with us today were former students of Dr. Caloy in his temporal bone lab Unfortunately, before finishing my residency training, Dr Caloy expeditiously left the PGH ENT department for unknown reasons. He then set up his private clinic in Quezon City and later joined the ENT department of University of Santo Tomas. Reflecting on the significant yet probably unknown achievements of Dr. Caloy toward the advancement of otology and neuro-otology in our country, I realize how blessed I was to be one of his students during that brief period when he was still with us at UP-PGH. With our profound gratitude Sir, we will always remember you.

The Base Radicals in Normal Classes of Complete Algebra

2021 ◽  
Vol 11 (12) ◽  
pp. 2012-2017
Author(s):  
宗文 杨
Keyword(s):  
Complete Algebra

Language Antinomies Concerning Spelling of Short Forms of Verbal Adjectives with н or нн

2021 ◽  
Vol 80 (4) ◽  
pp. 86
Author(s):  
Elena V. Beshcenkova
Keyword(s):  
Model Management ◽  
Distinctive Features ◽  
Short Forms ◽  
Quantitative Ratio ◽  
Grammatical Categories ◽  
Usual Norm ◽  
Google Books

According to the Russian orthography, the spelling of short forms of verbal words in -нный with н (n) or with нн (nn) before a non-zero ending depends on the grammatical status of the word: participle or adjective. It is believed that writing expresses a systemic opposition of grammatical categories, i.e. the reflective vector of the antinomy “reflective – conditional” operates in this area. But is it possible to check whether the antinomy was chosen correctly to describe a particular picture of the studied area of writing? Are there any distinctive features of the types of spellings obtained as a result of the action of one or another antinomy? We attempt to answer the latter using the example of the spelling of short verbal adjectives. Proceeding from the fact that short forms of participles are always written with only one letter н, and short forms of adjectives with a non-zero ending are threefold: either with нн, or with н, or both with н and with нн, depending on the meaning and / or model management, — we will evaluate the quantitative ratio of these groups, how they are recorded in different dictionaries, compare the codification and the usual norm, as reflected in the texts of the RNC resources and Google books, we will determine what is being implemented in writing steadily and what is not. Based on the results obtained, one can try to understand whether the letter regularly reflects precisely the grammatical opposition.

scholarly journals Automatic continuity for Banach algebras with finite-dimensional radical

2006 ◽  
Vol 81 (2) ◽  
pp. 279-296 ◽  
Author(s):  
Hung Le Pham
Keyword(s):  
Banach Algebra ◽  
Banach Algebras ◽  
Automatic Continuity ◽  
Algebraic Condition ◽  
Finite Dimensional ◽  
Complete Algebra

AbstractThe paper [3] proved a necessary algebraic condition for a Banach algebra A with finite-dimensional radical R to have a unique complete (algebra) norm, and conjectured that this condition is also sufficient. We extend the above theorem. The conjecture is confirmed in the case where A is separable and A/R is commutative, but is shown to fail in general. Similar questions for derivations are discussed.

scholarly journals An extension of a Hardy-Littlewood-Pólya inequality

1978 ◽  
Vol 21 (1) ◽  
pp. 11-15 ◽  
Author(s):  
A. Erdélyi
Keyword(s):  
Integrable Function ◽  
Right Hand ◽  
Usual Norm ◽  
Locally Integrable Function ◽  
Image Position ◽  
The Right

The Hardy-Littlewood-Pólya inequality in question can be written in the formHere and throughout, all functions are assumed to be locally integrable on ]0,∞[, 1≤p≤∞,p-1+(p′)-1=1 (with similar conventions for q,r,s), is the usual norm on Lp(0,∞), and if the right hand side is finite, then (1.1) is understood to mean thatdefines a locally integrable function Kf for which (1.1) holds.

scholarly journals Angle in the space of $ p $-summable sequences

2021 ◽  
Vol 7 (2) ◽  
pp. 2810-2819
Author(s):  
Muh Nur ◽  
◽  
Moch Idris ◽  
Firman ◽  
Keyword(s):  
Dimensional Subspace ◽  
Inner Product ◽  
Usual Norm

<abstract><p>The aim of this paper is to investigate completness of $ A $ that equipped with usual norm on $ p $-summable sequences space where $ A $ is subspace in $ p $-summable sequences space and $ 1\le p &lt; \infty $. We also introduce a new inner product on $ A $ and prove completness of $ A $ using a new norm that corresponds this new inner product. Moreover, we discuss the angle between two vectors and two subspaces in $ A $. In particular, we discuss the angle between $ 1 $-dimensional subspace and $ (s-1) $-dimensional subspace where $ s\ge 2 $ of $ A $.</p></abstract>

An Lp Saturation Theorem for Splines

1972 ◽  
Vol 24 (5) ◽  
pp. 957-966 ◽  
Author(s):  
G. J. Butler ◽  
F. B. Richards
Keyword(s):  
Lebesgue Measure ◽  
Measure Zero ◽  
Lebesgue Measure Zero ◽  
Usual Norm ◽  
Saturation Theorem ◽  
Image Position ◽  
Class Of Functions

Let 1 be a subdivision of [0, 1], and let denote the class of functions whose restriction to each sub-interval is a polynomial of degree at most k. Gaier [1] has shown that for uniform subdivisions △n (that is, subdivisions for which if and only if f is a polynomial of degree at most k. Here, and subsequently, denotes the usual norm in Lp[0, 1], 1 ≦ p ≦ ∞, and we should emphasize that functions differing only on a set of Lebesgue measure zero are identified.

Some James numbers of Stiefel manifolds

1982 ◽  
Vol 92 (1) ◽  
pp. 139-161 ◽  
Author(s):  
Hideaki Ōshima
Keyword(s):  
The Other ◽  
Stiefel Manifold ◽  
Complex Numbers ◽  
Real Numbers ◽  
The Real ◽  
Stiefel Manifolds ◽  
Other Hand ◽  
Usual Norm

The purpose of this note is to determine some unstable James numbers of Stiefel manifolds. We denote the real numbers by R, the complex numbers by C, and the quaternions by H. Let F be one of these fields with the usual norm, and d = dimRF. Let On, k = On, k(F) be the Stiefel manifold of all orthonormal k–frames in Fn, and q: On, k → Sdn−1 the bundle projection which associates with each frame its last vector. Then the James number O{n, k} = OF{n, k} is defined as the index of q* πdn−1(On, k) in πdn−1(Sdn−1). We already know when O{n, k} is 1 (cf. (1), (2), (3), (13), (33)), and also the value of OK{n, k} (cf. (1), (13), (15), (34)). In this note we shall consider the complex and quaternionic cases. For earlier work see (11), (17), (23), (27), (29), (31) and (32). In (27) we defined the stable James number , which was a divisor of O{n, k}. Following James we shall use the notations X{n, k}, Xs{n, k}, W{n, k} and Ws{n, k} instead of OH{n, k}, , Oc{n, k} and respectively. In (27) we noticed that O{n, k} = Os{n, k} if n ≥ 2k– 1, and determined Xs{n, k} for 1 ≤ k ≤ 4, and also Ws{n, k} for 1 ≤ k ≤ 8. On the other hand Sigrist (31) calculated W{n, k} for 1 ≤ k ≤ 4. He informed the author that W{6,4} was not 4 but 8. Since Ws{6,4} = 4 (cf. § 5 below) this yields that the unstable James number does not equal the stable one in general.

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