To do some other answers / comments, if you want CenterDot
to CenterDot
interpreted by both Times
as input and output, using something like
Unprotect[CenterDot, Times]; CenterDot = Times; Times /: MakeBoxes[Times[a__], fmt_] := With[{cbox = ToBoxes[HoldForm[CenterDot[a]]]}, InterpretationBox[cbox, Times[a]]]; Protect[CenterDot, Times];
What you can add to your init.m if you want it to be loaded by default.
This works with both numeric and symbolic expressions, for example
In[5]:= 1\[CenterDot]2\[CenterDot]3 Out[5]= 6 In[6]:= abc Out[6]= a\[CenterDot]b\[CenterDot]c
You can also make an automatically inserted multiplication character between numbers separated by spaces, CenterDot
by doing
SetOptions[EvaluationNotebook[], {AutoMultiplicationSymbol -> True, NumberMultiplier -> "\[CenterDot]"}]
or by choosing Dot Center in the Appearance> Numbers> Multiplication settings dialog box.
For instance:

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