Questions tagged [baer-rings]

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Algebraic language of fundamental results in operator algebras

Ignoring topological structures of von Neumann algebras, the larger category of Baer $*$-rings was emerged. In the unique text written by Sterling K. Berberian (1), some concepts and results in von ...
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38 views

Relation between left projections

Let $A$ be a Baer *-ring. Let $x$ be in $A$, $L(x)$ is the left projection of $x$ that is the smallest projection with $L(x)x=x$. Q. Let $p,q$ are projections in $A$ with $p\leq q$. For a given ...
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35 views

The statue of a sequence of finite projections

Let $A$ be a Baer $*$-ring. Let $\{p_n\}$ be a sequence of finite projections in $A$. True or false? Suppose that there is no $N$ with $p_n=p_{n+1}$ for $n\geq N$. We have then $\inf_{1\leq n\leq ...
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34 views

something concerning finite projections

Let $A$ be a Baer *-ring. Let $x$ be an isometry (meaning $x^*x=1$ where $1$ is the unit of $A$). Let $e$ be a finite projection in $A$ such that $ex^ne=ex^n$ for every $n\geq0$. Q. Can we say that ...
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2answers
109 views

Upper triangular $2\times2$-matrices over a Baer *-ring

Let $A$ be a Baer $*$-ring. Let us denote $B$ by the space of all upper triangular matrices $\left(\begin{array}{cc} a_1& a_2 \\ 0 & a_4 \end{array}\right)$ where $a_i$'s are in $A$. Is $B$ ...
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1answer
71 views

Strongly finite projections in $*$-rings

Let $A$ be a $*$-ring. Let us have some points: i) We recall that a projection $p$ is a self-adjoint idempotent that is $p=p^*=p^2$. ii) On the set of projections, we write $p\leq q$ if $pq=p$. iii)...
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66 views

Transmission of finite projections

Let $A$ be a Baer*-ring. Let us denote $L(x)$ by the left projection of $x$ (the smallest projection with $L(x)x=x$). Let $p$ be a finite projection in $A$. Is $L(xp)$ a finite projection for every $...
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141 views

Algebraic version of unilateral shift

It was confirmed that Wold-type decomposition can be extended from von Neumann algebras to Baer*-rings (see this paper). By algebraic tools the notion of unilateral shifts is successfully transmitted ...
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109 views

shifts in Baer*-rings

Let $A$ be a Baer*-ring with the unit $1$. An important point that holds in every Baer*-ring is this: for every element $y\in A$ there is an smallest projection $e_y\in A$, called the left projection ...
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106 views

When does a projection commute with an element?

Let $A$ be a Baer*-ring and $a$ be in $A$. Assume that for a projection $p\in A$, the following system of equations hold: \begin{array}{ll} a^na^{*n}p=p , & n\geq0\\ a^{*n}a^np=p , &...
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1answer
101 views

Is $M_n(Z_p)$ a Baer*-ring?

Let $p$ be a prime number and $n$ be a natural number. Does $M_n({\mathbf{Z}}_p)$, the ring of all $n\times n$ matrices over the field ${\mathbf{Z}}_p$ form a Baer*-ring? If not, what about for ...
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71 views

Does the inequality $x^2\leq x$ hold in Baer*-ring for $0 \leq x \leq 1$?

Let $A$ be a Baer*-ring and $x$ be a positive element with $0\leq x\leq1$ where $1$ is just the unit of $A$. Can we conclude $x^2\leq x$?
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1answer
154 views

A Baer *-ring which is not embedded into $B(H)$

Assume $A$ is a complex $*$-algebra which is also a Baer*-ring. Q. Can we concluded that there exists a Hilbert space $H$ such that $A$ is embedded in $B(H)$ as a Baer*-ring? What about when $A$ ...
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58 views

square root of contractions in Baer *-rings

Let $A$ be a unital Baer*-ring. We say that $a$ is a contraction if $aa^*\leq1$ and $a^*a\leq1$. Q1) Assume $a$ is a contraction. Has the positive element $1-aa^*$ any square root? (if yes, seems $1-...
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3answers
182 views

Some non-trivial Baer *-rings

A Baer *-ring is an *-algebra whose lattice of projections is complete. I know two well-handed kinds of these structures: 1- W*-algebras (abstract case of von Neumann algebras). 2- The inverse ...
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1answer
78 views

Two points concerning Baer *-rings

Let $A$ be a unital Baer *-ring. 1- Assume that $\{p_i\}$ is a family of projections in $A$. Let $x$ be an isometry in A (I mean $x^*x=1$ where $1$ is the unit of $A$). True or false: $\inf (...
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1answer
166 views

Normal reduced rings

It is known that a reduced commutative unitary Baer ring $A$ is normal iff for every prime ideal $\mathfrak p$ of $T(A)$ (here $T(A)$ is the total quotient of $A$) one has $A/(\mathfrak p\cap A)$ is ...