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### On Idempotent D-Norms [article]

Michael Falk
2014 arXiv   pre-print
If this iteration is repeatedly done on the same D-norm, then the limit of the track is idempotent.  ...  We characterize the set of idempotent D-norms. Iterating the multiplication provides a track of D-norms, whose limit exists and is again a D-norm.  ...  Then the limit lim n→∞ x n i=1 D (i) =: x D * , x ∈ R d , is an idempotent D-norm on R d . Proof. We know from Poposition 4.2 that · D * is a D-norm on R d .  ...

### On the norm of an idempotent Schur multiplier on the Schatten class

William D. Banks, Asma Harcharras
2004 Proceedings of the American Mathematical Society
We also give a simple characterization of those idempotent Schur multipliers on S p whose norm is 1.  ...  We show that if the norm of an idempotent Schur multiplier on the Schatten class S p lies sufficiently close to 1, then it is necessarily equal to 1.  ...  Introduction We study norms of idempotent Schur multipliers defined on the Schatten p-class with 1 < p < ∞, p = 2.  ...

### Flat ideals in the unit interval with the canonical fuzzy order [article]

Hongliang Lai, Dexue Zhang, Gao Zhang
2018 arXiv   pre-print
A characterization of flat ideals in the unit interval with the canonical fuzzy order is obtained with the help of the ordinal sum decomposition of continuous t-norms.  ...  idempotent then c = c + = c − . • For each non-idempotent element c, the restriction of & on [c − , c + ] is either isomorphic to the Lukasiewicz t-norm or to the product t-norm.  ...  As an immediate corollary of the above proposition we obtain that if & is a continuous t-norm and a, b (a < b) are idempotent elements of &, then the restriction of & on [a, b] is a continuous t-norm on  ...

### Page 73 of Mathematics Magazine Vol. 12, Issue 2 [page]

1937 Mathematics Magazine
As above but with (c+e)(d+/f)=1 The nilpotent is —(c+eitj III. One idempotent, two nilpotents. 0 ci+dj et+fj 0 The idempotent is (c+e)i+(d+f\j (c+e)(d+f)  ...  Three idempotents, no nilpotents in the algebra. t ci+dj eit+fj j c+exlxd-+f, (c+e)(d+f)#1 The third idempotent is (l—c—e)i+(1—d—f)j 1—(c+e)(d+f) II. Two idempotents, one nilpotent.  ...

### Page 73 of Mathematics Magazine Vol. 12, Issue 2 [page]

1937 Mathematics Magazine
As above but with (c+e)(d+f) =1 The nilpotent is —(c+e)i+j III. One idempotent, two nilpotents. 0 ci+dj ei+fj 0 The idempotent is (c+e)i+(d+f)j (c+e)(d+f)  ...  Three idempotents, no nilpotents in the algebra. t ci+dj ei +f j c+ex1lxd-f, (ct+e)(d+f)#1 The third idempotent is (l—c—e)t+(1—d—f)j 1—(c+e)(d+f) II. Two idempotents, one nilpotent.  ...

### A note on reduced Jordan algebras

Kevin McCrimmon
1968 Proceedings of the American Mathematical Society
The set of values QoiAa) = -Qe2,e,iA23) of Qo on AM is called the norm class of ei, and is denoted by A(eO.  ...  Consequently, Witt's theorem implies that the norm forms A and N' of D and D' are similar. Then D and D' are known to be isomorphic (see[l]).  ...

### A Note on Reduced Jordan Algebras

Kevin McCrimmon
1968 Proceedings of the American Mathematical Society
The set of values QoiAa) = -Qe2,e,iA23) of Qo on AM is called the norm class of ei, and is denoted by A(eO.  ...  Consequently, Witt's theorem implies that the norm forms A and N' of D and D' are similar. Then D and D' are known to be isomorphic (see[l]).  ...

### A note on involution rings

D. I. C. Mendes
2009 Miskolc Mathematical Notes
As an application, von Neumann regular involution rings with central idempotent norm elements are described.  ...  The structure of certain involution rings in which the norms are multiplicatively generated by nilpotents is also determined.  ...  If R is a von Neumann regular involution ring with central idempotent norm elements, then R is isomorphic to a subdirect sum of involution rings R i .i 2 /, where each R i is one of the following: (i)  ...

### Locally \$B\sp{\ast} \$-equivalent algebras

Bruce A. Barnes
1972 Transactions of the American Mathematical Society
Therefore |-|2 is a complete norm on (l-e)Af. But eAf is finite d;r. ^nsional, hence complete. Therefore Af=eAf® (1 -e)Af is si Hubert space. We denote the Hubert space Af by Jif.  ...  Assume that D is a *-subalgebra of A, and that there exists K> 0 such that Kv(t) ä || 11| for all t e Ds. Then D is B*-equivalent. Proof. * is symmetric on A by Proposition 2.1 (3) .  ...

### Norm one idempotent cb-multipliers with applications to the Fourier algebra in the cb-multiplier norm [article]

Brian E. Forrest, Volker Runde
2008 arXiv   pre-print
We characterize the norm one idempotents in M_cbA(G): the indicator function of a set E ⊂ G is a norm one idempotent in M_cbA(G) if and only if E is a coset of an open subgroup of G.  ...  (We can even slightly relax the norm bounds.)  ...  Hence, every norm one idempotent in M cb A(G) is a norm one idempotent in MA(G). By [Boż] , M cb A(F 2 ) MA(F 2 ) holds, and-as M.  ...

### Regular left-continuous t-norms

Thomas Vetterlein
2008 Semigroup Forum
The next part is preparatory; we discuss certain systems of continuous and idempotent functions (Section 5).  ...  The t-norm algebras based on regular l.-c. t-norms generate the variety of MTL-algebras. With each regular l.  ...  -c. t-norm. Then we call ([0, 1] ; ≤, , 0, 1) the t-norm monoid based on .  ...

### Classification of L^p AF algebras [article]

N. Christopher Phillips, Maria Grazia Viola
2017 arXiv   pre-print
As background, we develop the theory of matrix normed L^p operator algebras, and show that there is a unique way to make a spatial L^p AF algebra into a matrix normed L^p operator algebra.  ...  groups. 2) A B as rings. 3) A B (not necessarily isometrically) as Banach algebras. 4) A is isometrically isomorphic to B as Banach algebras. 5) A is completely isometrically isomorphic to B as matrix normed  ...  We don't define hermitian idempotents in a nonunital Banach algebra, since whether an idempotent is hermitian depends on the norm used on the unitization, even for L p operator algebras, as is shown by  ...

### Idempotents in group algebras

Walter Rudin
1963 Bulletin of the American Mathematical Society
(D) If ƒ is an idempotent on an abelian group and if ||/|| > 1, then ||/||^>!\/5 [3, p. 72]. (Note that there are no idempotents ƒ with jj/ll <1, except ƒ =0.)  ...  Let us draw attention to the following facts, of which (A) and (D) are probably the most striking: (A) If ƒ is an idempotent on an abelian group G, then the support group of f is finite.  ...

### Basic fuzzy logic and BL-algebras

P. Hájek
1998 Soft Computing - A Fusion of Foundations, Methodologies and Applications
by c o n tinuous t-norms.  ...  In Sect. 4 w e d e v elop some algebra of linearly ordered BL-algebras. In Sect. 5 we exhibit two additional axioms (B1), (B2) and show soundness and completeness of BL + (B1) + (B2) for t-algebras.  ...  Continuous t-norms We recall some well-known facts on continuous t-norms. A t-norm is a binary operation on 0 1] (i.e. t : 0 1] 2 !  ...

### Idempotent, model, and Toeplitz operators attaining their norms [article]

Neeru Bala, Kousik Dhara, Jaydeb Sarkar, Aryaman Sensarma
2021 arXiv   pre-print
We study idempotent, model, and Toeplitz operators that attain the norm.  ...  Here S_𝒬 = P_𝒬M_z|_𝒬, the compression of the shift M_z on the Hardy space H^2(𝔻) to 𝒬.  ...  Idempotent Operators It is evident that any orthogonal projection on a Hilbert space is norm attaining. Here we deal with the issue of norm attainment of idempotent operators.  ...
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